Answered

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The principal represents an amount of money deposited in a savings account subject to compound interest at the given rate.

| Principal | Rate | Compounded | Time |
|-----------|------|------------|------|
| [tex]$8000 | 4% | Monthly | 1 year|

A. Find how much money there will be in the account after the given number of years.
B. Find the interest earned.

\ \textless \ strong\ \textgreater \ Finance Formulas:\ \textless \ /strong\ \textgreater \

- \( A = P \left(1 + \frac{r}{n}\right)^{nt} \)
- \( P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \)
- \( A = Pe^{rt} \)
- \( Y = \left(1 + \frac{r}{n}\right)^n - 1 \)

Where:
- \( A \) is the balance in the account after \( t \) years
- \( P \) is the principal investment
- \( r \) is the annual interest rate in decimal form
- \( n \) is the number of compounding periods per year
- \( Y \) is the investment's effective annual yield in decimal form

\ \textless \ strong\ \textgreater \ Questions:\ \textless \ /strong\ \textgreater \

A. The amount of money in the account after 3 years is $[/tex] _______. (Round to the nearest hundredth as needed.)
B. The amount of interest earned is $ _______. (Round to the nearest hundredth as needed.)

Sagot :

GinnyAnswer
Let's solve the given problem step by step.

### Given Data:
- Principal ([tex]\( P \)[/tex]): [tex]$8000 - Annual Interest Rate (\( r \)): 4% or 0.04 (in decimal form) - Number of times the interest is compounded per year (\( n \)): 12 (monthly) - Time (\( t \)): 1 year ### Part A: Find how much money there will be in the account after the given number of years \( t = 1 \). First, we use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{n t} \] Substitute the values into the formula: \[ A = 8000 \left(1 + \frac{0.04}{12}\right)^{12 \times 1} \] Now calculate the amount: \[ A = 8000 \left(1 + 0.0033333\right)^{12} \] \[ A = 8000 (1.0033333)^{12} \] \[ A = 8000 \cdot 1.040813 \] \[ A = 8325.93 \] So, the amount of money in the account after 1 year is \$[/tex]8325.93.

### Part B: Find the interest earned.

The interest earned can be calculated by subtracting the principal from the total amount after 1 year.

[tex]\[ \text{Interest Earned} = A - P \][/tex]

Substitute the values:
[tex]\[ \text{Interest Earned} = 8325.93 - 8000 \][/tex]
[tex]\[ \text{Interest Earned} = 325.93 \][/tex]

So, the amount of interest earned is \[tex]$325.93. ### Summary: A. The amount of money in the account after 1 year is \$[/tex]8325.93.\
B. The amount of interest earned is \$325.93.
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